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The problem asks us to determine the equation of a line that passes through the point \((4, -2)\) and is perpendicular to the line given by the equation \(y = 2x + 2\).

### Solution:
1. **Identify the slope of the given line**: 
   - The given line is \(y = 2x + 2\). The slope (\(m\)) of this line is \(2\).

2. **Find the slope of the perpendicular line**: 
   - For two lines to be perpendicular, the product of their slopes must be \(-1\). If the slope of the given line is \(2\), then the slope of the perpendicular line (\(m_{\perp}\)) is:
     \[
     m_{\perp} = -\frac{1}{2}
     \]

3. **Use the point-slope form**:
   - The point-slope form of a line is:
     \[
     y - y_1 = m(x - x_1)
     \]
     Substituting the point \((4, -2)\) and \(m = -\frac{1}{2}\):
     \[
     y - (-2) = -\frac{1}{2}(x - 4)
     \]
     Simplifying this, we get:
     \[
     y + 2 = -\frac{1}{2}(x - 4)
     \]

4. **Simplify the equation**:
   - Distribute \(-\frac{1}{2}\):
     \[
     y + 2 = -\frac{1}{2}x + 2
     \]
   - Subtract \(2\) from both sides:
     \[
     y = -\frac{1}{2}x
     \]

### Final Answer:
The equation of the line that passes through \((4, -2)\) and is perpendicular to \(y = 2x + 2\) is:
\[
y = -\frac{1}{2}x
\]

Would you like a detailed explanation of any step, or do you have other questions?

### Related Questions:
1. What is the general form of a linear equation?
2. How do you find the slope of a line given two points?
3. What is the difference between parallel and perpendicular lines in terms of their slopes?
4. How do you convert point-slope form to slope-intercept form?
5. What would be the equation of a line parallel to \(y = 2x + 2\) through the point \((4, -2)\)?

### Tip:
For perpendicular lines, remember that if one line's slope is \(m\), the slope of the line perpendicular to it will always be \(-\frac{1}{m}\).

Determine the equation of the line that passes through point (4, -2) and is perpendicular to the line y = 2x + 2.

Learn how to find the equation of a line that is perpendicular to y = 2x + 2 and passes through the point (4, -2). This problem covers key algebraic concepts, including slopes, point-slope form, and perpendicular line slopes. Suitable for students in Grades 9-11.

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